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# continuous

Let $X$ and $Y$ be topological spaces. A function $f\colon X\to Y$ is continuous if, for every open set $U\subset Y$, the inverse image $f^{{-1}}(U)$ is an open subset of $X$.

In the case where $X$ and $Y$ are metric spaces (e.g. Euclidean space, or the space of real numbers), a function $f\colon X\to Y$ is continuous if and only if for every $x\in X$ and every real number $\epsilon>0$, there exists a real number $\delta>0$ such that whenever a point $z\in X$ has distance less than $\delta$ to $x$, the point $f(z)\in Y$ has distance less than $\epsilon$ to $f(x)$.

Continuity at a point

A related notion is that of local continuity, or continuity at a point (as opposed to the whole space $X$ at once). When $X$ and $Y$ are topological spaces, we say $f$ is continuous at a point $x\in X$ if, for every open subset $V\subset Y$ containing $f(x)$, there is an open subset $U\subset X$ containing $x$ whose image $f(U)$ is contained in $V$. Of course, the function $f\colon X\to Y$ is continuous in the first sense if and only if $f$ is continuous at every point $x\in X$ in the second sense (for students who haven’t seen this before, proving it is a worthwhile exercise).

In the common case where $X$ and $Y$ are metric spaces (e.g., Euclidean spaces), a function $f$ is continuous at $x\in X$ if and only if for every real number $\epsilon>0$, there exists a real number $\delta>0$ satisfying the property that $d_{Y}(f(x),f(z))<\epsilon$ for all $z\in X$ with $d_{X}(x,z)<\delta$. Alternatively, the function $f$ is continuous at $a\in X$ if and only if the limit of $f(x)$ as $x\to a$ satisfies the equation

$\lim_{{x\to a}}f(x)=f(a).$ |

## Mathematics Subject Classification

26A15*no label found*54C05

*no label found*81-00

*no label found*82-00

*no label found*83-00

*no label found*46L05

*no label found*

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## Attached Articles

If $f\colon X\to Y$ is continuous then $f\colon X\to f(X)$ is continuous by matte

composition of continuous mappings is continuous by mathcam

continuity is preserved when codomain is extended by matte

gluing together continuous functions by yark

equivalent formulations for continuity by matte

continuity of natural power by pahio

sequentially continuous by ehremo

graph theorems for topological spaces by joking

continuity of sine and cosine by pahio

## Comments

## metric spaces

would it be possible to have a more elementary definition that doesn't involve metric spaces ?

## Re: metric spaces

It is probably not possible to have a more elementary definition by omitting metric spaces. The most elementary possible definitions of continuous (that is, the ones you see in first year calculus texts) still need to use deltas and epsilons, and given that a correct definition needs to use deltas and epsilons, using the language of metric spaces adds little extra complexity.

The correct question, perhaps, is whether it is possible to give a more elementary definition not involving _topological_ spaces. I attempted to do just that in the second paragraph.

Some have the opinion that the nonstandard analysis definition of continuous (namely, a function is continuous if and only if infinitesimally close points in the domain have infinitesimally close images) is the simplest of all, but the nonstandard analysis approach has the drawback that you must define infinitesimals, and I think that no net effort is saved in the end by going this route.

## non-standard definition

I agree completely with you that the non-standard definition is not shorter or simpler once you take into account the fact that, to use it you first have to define non-standard numbers. To do that either reequires ultrafilers or model theory, and both of these are advanced topics.

However, I still thiink it would be a good idea to add the non-standard definition. My reason for saying this is that, if this is supposed to be an encyclopaedia, it should try to include everything, easy or hard. Maybe you could add a sentence like "In non-standard analysis, a function is defined to be continuous if ...". One of these days (if someone else doesn't do it first), I plan to add something about non-standard analysis. Probably, I'll lump it together with p-adic analysis in a topic entry on non-Archimedian analysis.

## inverse image of _any_ U in Y?

I think there is some sort of subtlety I'm missing, because it seems like the following example would be non-continuous under the current definition:

X = [0,1]

Y = Real numbers

f(x) = 0 (constant for all x in [0,1]).

Then if U = (-1,1), U is open and in Y, but yet F^-1(U) = {0} which is not open.

A constant function is not continuous!? What is wrong?

## Re: inverse image of _any_ U in Y?

Whoops, means to say that F^-1(U) = [0,1]. Same problem still applies.

## Re: inverse image of _any_ U in Y?

The set [0,1] is not open in the topology of R, but is open in the topology induced by (the Euclidean topology of) R on it.

And you moust think f as a function from the topological space [0,1] with the topology induced by (the Euclidean topology of) R, to the topological space R with the Euclidean topology.

Then f^-1((-1,1)) is the whole set [0,1]---which *is* open.

## Re: inverse image of _any_ U in Y?

I'll try to give you a hint:

f must be a function between topological spaces. How is X=[0,1] a topological space? I mean, what topology were you thinking about when you picked it?

I'm sure you meant the topology inherited by the topology of Real numbers (search for "subspace topology", in case you are not familiar with it).

In this case, [0,1] is indeed an open set of X and your function is indeed continuous.. (Of course, this doesn't mean that [0,1] is an open set in R).

Think about it!

## Re: inverse image of _any_ U in Y?

Oh wow, tricky how that works. Thanks all.